Cartan-Hadamard theorem for metric spaces

1. Introduction

Contrary to its name, the Cartan-Hadamard theorem originates in the work of Hans von Mangoldt. He proved in 1881 that two arbitrarily close geodesics on a real analytic surface of nonpositive curvature can intersect at most once. Jacques Hadamard then developed this theory in 1898 by showing that each homotopy class of paths connecting two points contains a unique geodesic. Both of these results can be found on page 50 of [6]. Élie Cartan later generalized Hadamard’s theory to a wider class of manifolds in [3], giving us the name Cartan-Hadamard.

In its more modern form, the Cartan-Hadamard theorem is a statement about universal coverings. This is because, for a certain class of topological spaces, a universal cover can be constructed via homotopy classes of paths starting at a point. Hence, in the case of a complete, connected Riemannian manifold of nonpositive curvature, our homotopy classes of paths are represented by geodesics starting at a point; in particular, our covering map is the exponential map, cf. [2]. This theory was later generalized by M. Gromov, [5], then S. Alexander and R. Bishop to complete geodesic spaces with a locally convex metric (locally m-convex), [1].

2. Covering spaces

Let $ A, E $ be topological spaces, and let $ \pi \colon E \to A $ be a surjective continuous map. Suppose that for each $ a \in A $ there exists an open neighborhood $ U $ and a decomposition

\[\pi^{-1}(U)=\coprod_{i\in E_a}V_i,\]

where the $ V_i $ are pairwise disjoint open subsets of $ E $ and each restriction $ \pi\mid_{V_i}\colon V_i\to U $ is a homeomorphism. Then $ \pi \colon E \to A $ is called a covering. A connected covering with simply connected total space is called universal. If basepoints $ e_0\in E $ and $ f_0\in F $ lie over the same point of $ A $, then a universal cover $ \pi\colon E\to A $ has the following property: for any connected covering $ \varphi\colon F\to A $, there is a unique map $ h\colon E\to F $ such that $ \varphi\circ h=\pi $ and $ h(e_0)=f_0 $. In particular, if $ A $ is simply connected, then every connected covering of $ A $ is a homeomorphism.

We now construct the universal cover for a suitably nice class of topological spaces.

Theorem 2.1 Any connected, locally path connected, and semilocally simply connected topological space admits a universal cover.

Proof outline. Choose a basepoint $ a \in A $, and let $ E $ be the set of homotopy classes, relative to endpoints, of paths starting at $ a $. Choose path-connected open subsets $ U\subseteq A $ for which every loop in $ U $ is null-homotopic in $ A $. If $ \gamma(1)\in U $, let $ U_{[\gamma]} $ consist of the classes $ [\gamma\star\omega] $, where $ \omega $ is a path in $ U $ beginning at $ \gamma(1) $. These sets form a basis for a topology on $ E $, and the endpoint map

\[\pi([\gamma])=\gamma(1)\]

restricts to a homeomorphism $ U_{[\gamma]}\to U $. The usual path- and homotopy-lifting arguments show that $ E $ is simply connected and that $ \pi $ is universal. See [8], chapter 3, §8 for the complete proof. Q.E.D.

3. Metric space analogues

Let us recall the Riemannian

Cartan-Hadamard theorem. Let $ M $ be a complete, connected Riemannian manifold of nonpositive curvature. Fix a point $ p $. Then the exponential $ \text{exp}_p \colon T_p M \to M $ is a universal cover.

We would like to understand what it means for a metric space to have nonpositive curvature. For Riemannian manifolds, nonpositive sectional curvature is equivalent to both local m-convexity and locally being a $ \text{CAT}(0) $ space. For geodesic metric spaces, the latter condition implies the former, and therefore we will develop our theory in the more general locally m-convex case.

A real function $ f \colon \mathbb{R} \to \mathbb{R} $ is called convex if for every $ a, b \in \mathbb{R} $ and $ t \in [0, 1] $,

\[f(ta + (1-t)b) \leq t f(a) + (1-t) f(b).\]

In particular, if $ f $ is second-differentiable, then convexity equates to $ f''(x) $ being nonnegative. Let $ \gamma \colon [0, 1] \to X $ be a curve parameterized proportional to arc length. If $ \gamma $ is locally length minimizing between its endpoints, then it is called a geodesic. If the length of $ \gamma $ equals the distance between its endpoints, then it is called a minimizing geodesic. A metric space in which every two points can be connected by a (unique) minimizing geodesic is called a (unique) geodesic space. If any point has a neighborhood which is a (unique) geodesic space, then it is called a local (unique) geodesic space.

Let $ (X, d) $ be a geodesic space and $ f \colon X \to \mathbb{R} $. If for any minimizing geodesic $ \gamma $, the composition $ f \circ \gamma $ is a real convex function, then we call $ f $ a convex function. We call $ X $ m-convex if for every pair of minimizing geodesics $ \alpha,\beta\colon[0,1]\to X $ parameterized proportional to arc length, the function

\[t\longmapsto d(\alpha(t),\beta(t))\]

is convex. We call $ X $ locally m-convex if every point has a geodesic neighborhood on which this condition holds.

We also need a metric space analogue of the tangent space at a point. Fix a point $ x $ in a geodesic space $ X $. We denote by $ G_x $ the set of geodesics $ \gamma\colon[0,1]\to X $ starting at $ x $, and equip it with the uniform metric

\[\rho(\gamma,\eta)=\max_{0\leq t\leq 1}d(\gamma(t),\eta(t)).\]

We define the evaluation map $ \text{ev}_x \colon G_x \to X $ by $ \text{ev}_x(\gamma) = \gamma(1) $.

4. Main proof

We now follow the proof of Alexander and Bishop [1]. The most delicate point is local surjectivity of the evaluation map; we record the midpoint argument carefully enough to indicate where completeness and local m-convexity enter.

Proposition 4.1. Let $ X $ be a complete, locally m-convex geodesic space. Fix a point $ x $ in $ X $. Then $ \text{ev}_x \colon G_x \to X $ is a local isometry.

Proof outline. Fix $ \gamma\in G_x $. Compactness of $ \gamma([0,1]) $ and local m-convexity give $ r>0 $ such that all of the geodesic constructions below take place in m-convex neighborhoods of uniform radius $ r $. If two geodesics $ \alpha_1,\alpha_2 $ lie within uniform distance $ r $ of $ \gamma $, local convexity patches along their images to show that

\[\rho(\alpha_1,\alpha_2) =\max\{d(\alpha_1(0),\alpha_2(0)),d(\alpha_1(1),\alpha_2(1))\}.\]

In particular, for geodesics based at $ x $, their uniform distance equals the distance between their endpoints. It remains to prove that every endpoint sufficiently close to $ \gamma(1) $ is attained by a nearby geodesic.

For $ L>0 $, let $ P(L) $ be the following statement:

For every subsegment $ \overline{\gamma} $ of $ \gamma $ of length at most $ L $, any two points $ p,q $ whose respective distances from the endpoints of $ \overline{\gamma} $ are less than $ r/2 $ are joined by a unique geodesic $ \alpha(p,q) $ whose uniform distance from $ \overline{\gamma} $ is less than $ r/2 $.

The choice of $ r $ makes $ P(r) $ a local consequence of m-convexity. We claim that $ P(L) $ implies $ P(3L/2) $. Let $ p_0,q_0 $ trisect a subsegment of length at most $ 3L/2 $, and let $ p,q $ be the perturbed endpoints. Assuming $ P(L) $, define recursively

\[\begin{aligned} p_i &= \operatorname{Mid}(\alpha(p,q_{i-1})),\\ q_i &= \operatorname{Mid}(\alpha(p_{i-1},q)). \end{aligned}\]

Convexity of the distance between the relevant geodesics gives

\[d(p_{i-1},p_i)\leq \frac{R}{2^i}, \qquad d(q_{i-1},q_i)\leq \frac{R}{2^i},\]

where $ R<r/2 $ bounds the endpoint perturbations. Thus $ (p_i) $ and $ (q_i) $ are Cauchy. Completeness gives limits $ p_\infty,q_\infty $, while m-convexity gives uniform convergence of the associated geodesics. The two limiting geodesics overlap along the unique geodesic from $ p_\infty $ to $ q_\infty $ and therefore glue to a geodesic from $ p $ to $ q $. This proves $ P(3L/2) $.

Iterating the implication proves $ P(L) $ for every $ L $. Taking the left endpoint to be $ x $ shows that $ \text{ev}_x $ maps a uniform ball about $ \gamma $ onto a metric ball about $ \gamma(1) $. The endpoint-distance equality above shows that this map is an isometry. For the full details of the choice of $ r $ and the limiting argument, see the proof of Theorem 2 in [1]. Q.E.D.

We next recall the path-lifting fact used to turn a complete local isometry into a covering.

Lemma 4.2. Let $ f \colon Y \to X $ be a local isometry with $ Y $ complete. Let $ x \in X $ and $ y \in f^{-1}(x) $. Then every rectifiable curve $ \gamma\colon[0,1]\to X $ starting at $ x $ has a unique lift $ \omega\colon[0,1]\to Y $ starting at $ y $.

Proof. Local inverses give a unique lift on some initial interval. Let $ [0,T) $ be its maximal domain. Since a local isometry preserves the lengths of rectifiable curves, the diameter of the lifted tail $ \omega([s,t]) $ is at most the length of $ \gamma\mid_{[s,t]} $. If $ T<1 $, rectifiability implies that $ \omega(t) $ is Cauchy as $ t\to T $. Completeness supplies a limit $ y_T $, and a local isometry neighborhood of $ y_T $ extends the lift past $ T $, contradicting maximality. Hence $ T=1 $. Uniqueness follows from uniqueness in local isometry neighborhoods and a standard open-and-closed argument. Q.E.D.

Proposition 4.3. Let $ \pi \colon \widetilde{X} \to X $ be a local isometry between nonempty complete length spaces. Suppose every point of $ X $ has a neighborhood of bipoint uniqueness: any two points in the neighborhood are joined by a unique minimizing geodesic, and these geodesics vary continuously with their endpoints. Then $ \pi $ is a covering.

Proof outline. First, $\pi$ is surjective. Indeed, start at any $\widetilde x_0\in\widetilde X$. Because $X$ is a length space, $\pi(\widetilde x_0)$ can be joined to any $x\in X$ by a rectifiable curve, and Lemma 4.2 lifts that curve to a path whose endpoint lies over $x$.

Choose a sufficiently small ball $U=B(x,\varepsilon)$ contained in a neighborhood of bipoint uniqueness. For each $y\in U$, let $\gamma_y$ be the unique minimizing geodesic from $x$ to $y$. Given $\widetilde x\in\pi^{-1}(x)$, Lemma 4.2 lifts $\gamma_y$ from $\widetilde x$; define $s_{\widetilde x}(y)$ to be its endpoint. Continuous dependence of $\gamma_y$ on $y$, together with a finite subdivision into local-isometry neighborhoods, shows that $s_{\widetilde x}$ is continuous. Moreover,

\[\pi\circ s_{\widetilde x}=\operatorname{Id}_U.\]

After decreasing $\varepsilon$ if necessary, uniqueness of local lifts makes $s_{\widetilde x}$ a local inverse to $\pi$, so its image is open and $\pi$ restricts there to a homeomorphism onto $U$. These images are pairwise disjoint: if two radial lifts ended at the same point, lifting the reversed radial geodesic would force their initial points over $x$ to agree. Finally, every point $\widetilde y\in\pi^{-1}(U)$ lies in one of the images. Indeed, lift the reversed geodesic from $\pi(\widetilde y)$ to $x$ beginning at $\widetilde y$; if its endpoint is $\widetilde x$, uniqueness says that reversing this lift is the radial lift defining $s_{\widetilde x}(\pi(\widetilde y))=\widetilde y$. Thus

\[\pi^{-1}(U)=\coprod_{\widetilde x\in\pi^{-1}(x)}s_{\widetilde x}(U),\]

so $U$ is evenly covered. This is the radial-lifting argument in Lemma 1 of [1]. Q.E.D.

We are now ready to state the main result with the global geodesic-space hypothesis used in [1].

Theorem 4.4. (Cartan-Hadamard) Let $ X $ be a complete, locally m-convex geodesic space. Fix a point $ x $ in $ X $. Then the evaluation map $ \text{ev}_x \colon G_x \to X $ is a universal cover. Consequently, every endpoint-preserving homotopy class of curves between two points in $ X $ contains exactly one geodesic.

Proof. Proposition 4.1 gives that $ \text{ev}_x $ is a local isometry. Local m-convexity gives neighborhoods of bipoint uniqueness: if two minimizing geodesics have the same endpoints, convexity forces the distance between corresponding points to vanish, and the same inequality gives continuous dependence on the endpoints.

The uniform metric $ \rho $ on $ G_x $ is complete. Indeed, a uniformly Cauchy sequence of geodesics converges pointwise in the complete space $ X $, and local m-convexity shows that the uniform limit is again a geodesic. The metric $ \rho $ need not be a length metric, so let $ \widehat{\rho} $ be its induced intrinsic metric. Proposition 4.1 implies that $ \rho $ and $ \widehat{\rho} $ agree locally and induce the same topology. Since $ \rho\leq\widehat{\rho} $, a $ \widehat{\rho} $-Cauchy sequence converges first in $ \rho $ and then, using local equality of the metrics, in $ \widehat{\rho} $. Thus $ (G_x,\widehat{\rho}) $ is a complete length space, and Proposition 4.3 shows that $ \text{ev}_x $ is a covering.

Finally, $ G_x $ is contractible: the homotopy

\[H(s,\gamma)(t)=\gamma(st)\]

contracts each geodesic through its initial subsegments to the constant geodesic at $ x $. Hence the covering is universal. The fiber over a point records endpoint-preserving homotopy classes of paths from $ x $, so each such class contains exactly one geodesic. Q.E.D.

5. References